Extrema of spectral band functions of two dimensional discrete periodic Schrödinger operators

TL;DR

The paper analyzes extrema of the spectral band functions $\lambda_j(k)$ for discrete periodic Schrödinger operators on $\mathbb{Z}^2$ by reducing to Floquet fibers $D_V(k)$ and studying the zeros of the associated Laurent polynomial $\mathcal{P}(z,\lambda)$. It refines the algebraic approach to bound level-set cardinalities by combining an improved Bézout bound (via a change of variables) with the Bernstein–Khovanskii–Kushnirenko bound, establishing a sharp bound of $4q_1q_2$ on the number of extrema-level points when $q_1$ and $q_2$ are coprime. The paper also derives intermediate Bézout-based bounds (e.g., $9q_1q_2-3$) to compare approaches. Together, these results connect spectral theory of discrete periodic operators with modern algebraic geometry, offering tighter constraints on the structure of extrema in spectral bands and informing the behavior of Bloch varieties for two-dimensional lattices.

Abstract

We use Bézout's theorem and Bernstein-Khovanskii-Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schrödinger operators on $\mathbb{Z}^2$. These approaches improve upon previous results of Liu and Filonov-Kachkovskiy.

Figures (1)

• Figure 1: $N$, a convex area containing all of the exponents of $\mathcal{P}(z,\lambda_{*})$.

Theorems & Definitions (25)

• Proposition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Definition 1
• Theorem 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• Remark 1